3.93 \(\int \log (\frac{c (b+a x)}{x}) \, dx\)

Optimal. Leaf size=25 \[ x \log \left (a c+\frac{b c}{x}\right )+\frac{b \log (a x+b)}{a} \]

[Out]

x*Log[a*c + (b*c)/x] + (b*Log[b + a*x])/a

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Rubi [A]  time = 0.0115837, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2453, 2448, 263, 31} \[ x \log \left (a c+\frac{b c}{x}\right )+\frac{b \log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Log[(c*(b + a*x))/x],x]

[Out]

x*Log[a*c + (b*c)/x] + (b*Log[b + a*x])/a

Rule 2453

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.), x_Symbol] :> Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /;
FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \log \left (\frac{c (b+a x)}{x}\right ) \, dx &=\int \log \left (a c+\frac{b c}{x}\right ) \, dx\\ &=x \log \left (a c+\frac{b c}{x}\right )+(b c) \int \frac{1}{\left (a c+\frac{b c}{x}\right ) x} \, dx\\ &=x \log \left (a c+\frac{b c}{x}\right )+(b c) \int \frac{1}{b c+a c x} \, dx\\ &=x \log \left (a c+\frac{b c}{x}\right )+\frac{b \log (b+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0048467, size = 28, normalized size = 1.12 \[ \frac{(a x+b) \log \left (\frac{c (a x+b)}{x}\right )}{a}+\frac{b \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(c*(b + a*x))/x],x]

[Out]

(b*Log[x])/a + ((b + a*x)*Log[(c*(b + a*x))/x])/a

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Maple [A]  time = 0.188, size = 44, normalized size = 1.8 \begin{align*} -{\frac{b}{a}\ln \left ({\frac{bc}{x}} \right ) }+x\ln \left ( ac+{\frac{bc}{x}} \right ) +{\frac{b}{a}\ln \left ( ac+{\frac{bc}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a*x+b)/x),x)

[Out]

-b/a*ln(b*c/x)+x*ln(a*c+b*c/x)+b*ln(a*c+b*c/x)/a

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Maxima [A]  time = 1.17715, size = 34, normalized size = 1.36 \begin{align*} x \log \left (\frac{{\left (a x + b\right )} c}{x}\right ) + \frac{b \log \left (a x + b\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a*x+b)/x),x, algorithm="maxima")

[Out]

x*log((a*x + b)*c/x) + b*log(a*x + b)/a

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Fricas [A]  time = 1.93737, size = 63, normalized size = 2.52 \begin{align*} \frac{a x \log \left (\frac{a c x + b c}{x}\right ) + b \log \left (a x + b\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a*x+b)/x),x, algorithm="fricas")

[Out]

(a*x*log((a*c*x + b*c)/x) + b*log(a*x + b))/a

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Sympy [A]  time = 0.341546, size = 20, normalized size = 0.8 \begin{align*} x \log{\left (\frac{c \left (a x + b\right )}{x} \right )} + \frac{b \log{\left (a x + b \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a*x+b)/x),x)

[Out]

x*log(c*(a*x + b)/x) + b*log(a*x + b)/a

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Giac [A]  time = 1.25673, size = 35, normalized size = 1.4 \begin{align*} x \log \left (\frac{{\left (a x + b\right )} c}{x}\right ) + \frac{b \log \left ({\left | a x + b \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a*x+b)/x),x, algorithm="giac")

[Out]

x*log((a*x + b)*c/x) + b*log(abs(a*x + b))/a